Theresa May with David Cameron, whom she will succeed as prime minister on Wednesday.
Photograph: Tim Rooke/Rex/Shutterstock

Theresa May will find a number of pressing matters in her in-tray on her first day in No 10 – and not just about how Britain will leave the EU. From universal credit to junior doctors’ strikes, school academisation and farmers’ subsidies, she will inherit many challenges from David Cameron’s administration.

The EU

Simply saying Brexit means Brexit, as May frequently does, leaves all the relevant questions unanswered. The German chancellor, Angela Merkel, will be wanting a clearer picture soon, including whether the UK is going to give up its presidency of the EU in the second half of next year.

The Cabinet Office minister Oliver Letwin and one of May’s favoured Home Office civil servants, Oliver Robbins, have already been appointed by Cameron to look through the options for withdrawal. May has said she will set up a Brexit department. It is quite likely some EU states – probably not Germany – will demand that she speed up her plan to delay triggering article 50, the clause that starts the two-year process of withdrawal, until the turn of the year.

The civil service is likely to tell her that she should not start the clock ticking until she knows what kind of relationship with the EU she wants. The simplest – but politically difficult – option is a version of Norway’s relationship with the EU. This would probably require accepting free movement, and payment of a fee for access to the single market. The more complex option is a unique relationship that ends free movement but retains some access to the single market. This will require a knowledge of trade negotiations that the UK lacks.

Eurosceptic MPs will want a pure sceptic in charge of the talks but that represents a diplomatic risk. It will take great skill to balance competing demands inside her party. Patrick Wintour

The economy

Stock markets crave political certainty and cheap money to invest. The elevation of May to No 10 secures the first. A vote by the Bank of England’s interest rate setting committee on Thursday to cut the 0.5% base rate and pump extra funds into the financial system will meet the second. The post-referendum jump in the FTSE 100 from below 6,000 to almost 6,700 shows investors are clearly confident that a cut in the cost of credit is on its way.

One reason the Bank should vote for a fresh stimulus is that the economy is clearly in need of help. The most recent industry surveys showed growth had slowed to 0.2% in the second quarter. Housebuilding has ground to a halt and last month consumer confidence dived at its fastest pace for 21 years. Businesses will be looking to May for an extended timetable for Brexit that prevents any further shocks. Phillip Inman

After six years of George Osborne in No 11, even City firms are coming round to the idea that austerity has run its course. With interest rates already at record lows, May’s chancellor may choose to borrow for spending on infrastructure to support business activity and jobs.

The UK has a record current account deficit, and the chancellor and governor of the Bank of England have both warned it has to be tackled. Five years after Osborne promised a “march of the makers”, Britain’s balance of payments is in a parlous state and manufacturing is struggling. The pound’s fall will not do much to help if exporters are fearful about post-Brexit trade barriers.

Then there’s the City. May needs a deal with the EU to deter banks and other financial firms from moving thousands of jobs to Europe – because 12% of GDP comes from financial services. Sean Farrell

She has said she is committed to UK membership of Nato and to the alliance’s target of spending 2% of GDP annually on defence.

Her first major test could come if there is a decision to be made on sending troops to help stabilise Libya. The UK has provisionally offered to send 800-1,300 soldiers to join an Italian-led force. Ewen MacAskill

Home affairs

They will also face delicate Whitehall negotiations over the government’s long-delayed legislation on extremism, including the issue of banning non-violent extremists without falling foul of freedom of speech judgments in the courts.

But it will be on the success or failure in curbing EU migration that the next home secretary will be judged. May has already set out some possibilities, including being much tougher on the 30% of labour migrants who arrive in Britain without a specific job. But moves against overseas students, including ensuring they leave the country at the end of their studies, can also be expected. Alan Travis

Foreign affairs

The Foreign Office is desperate to send out a message that the EU vote does not signal a more isolationist, inward-looking stance. Diplomats’ morale will only worsen if May does not tend to fractured relations by insisting the UK is engaged in Nato, the United Nations and the G20.

A UK-sponsored conference on Yemen later this month, including a visit by the US secretary of state, John Kerry, will be her first outing on the world stage. On 4 September she is due to attend the G20 in China, as well as the UN general assembly.

UK relations with Russia are at an all-time low, and the special relationship with China developed by the probably outgoing chancellor, George Osborne, will be in need of attention.

She will also be under pressure from the Tory press and grassroots to row back on Cameron’s cherished commitment to spend 0.7% of UK GDP on overseas aid. The pledge, a symbol of Cameron’s modernising credentials, will hold less totemic sway over May. Patrick Wintour

Health

The biggest short-term issue is the stand-off with junior doctors. Jeremy Hunt has said he will impose a new contract on them after it was rejected in a ballot of British Medical Association (BMA) members. Whether he stays on as health secretary could be a signal of May’s attitude to the dispute.

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More pressing in the long-term is the NHS budget. Hospitals in England ran up a record deficit of £2.45bn in 2015-16. Michael Gove promised an extra £100m a week for the NHS, which he said could be taken from Britain’s contributions to the EU. May has made no such pledge, and the BMA says even that would fill less than a quarter of the shortfall.

NHS bosses will also want urgent clarification on the status of EU nationals, with an estimated 40,000 working in the health service alone and more in the private care sector. Jessica Elgot

Education

For the last two years No 10 has driven the push for free schools and academies in England. May is unlikely to be any different. And British universities will expect little relief from the stringent visa regime on overseas students that May imposed as home secretary.

During a brief spell as Conservative shadow education secretary that ended in 2001, May backed the party orthodoxy in favour of grammar schools. That policy ended under Cameron and Gove, but she might revive increased academic selection. As recently as last year she backed an attempt to extend a grammar school in her constituency.

Two issues require immediate attention. One is the present and future status of EU nationals as pupils in British schools. The other is the “fair funding formula” for state schools in England, to even out funding between urban schools and the rest. May has supported it in the past, but unless there is new money involved the formula may disappoint many headteachers. Richard Adams

Despite these cuts, the government may struggle to keep down overall welfare spending as housing benefit claims continue to soar, and if the economy, as expected, experiences a Brexit-related downturn.

The biggest operational headache for the Department for Work and Pensions will continue to be the controversial, much-delayed universal credit programme. But the biggest political decision could be around pensions: after years of triple-locked inflation-busting annual rises for pensioners, will May dare target this most powerful of voting blocs for cuts? Patrick Butler

Scotland

Nicola Sturgeon has already indicated that she expects “early engagement” with the new prime minister over Brexit. Sturgeon has said she will continue to pursue every avenue possible to secure Scotland’s continued place within the EU.

Obviously one way of doing so is to hold a second referendum on independence. Although Ruth Davidson, a May supporter and the leader of the Scottish Conservatives was in London on Tuesday warning that such a move would be “irresponsible”, she has also said that a future Westminster government should not block Holyrood from holding another referendum. May – who campaigned for the union in the 2014 independence referendum – has yet to address the issue directly. But the potential breakup of the UK following the split with Europe must surely be high on her agenda. Libby Brooks

Northern Ireland

During the EU referendum campaign May warned that if there was a Brexit vote it was “inconceivable” that there would be no changes to the border between Northern Ireland and the Irish Republic. Now she will have to roll back from that prediction and assure the Irish government and nationalists in Northern Ireland that the border will not be fortified. Nationalists will be enraged if border posts or customs checks are reimposed.

May will also have to deal with demands from both communities for extra Treasury support to plug the EU funding gap. An estimated £8.20 of every £10 of Ulster farming income comes from the EU. She will also face demands from Sinn Féin for a referendum on Northern Ireland’s place within the UK, which will intensify if Scotland goes for another independence vote. Henry McDonald

Media

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The BBC has agreed in principle terms for renewing its royal charter to 2028, which include taking on the £700m-plus cost of free TV licences for over-75s, and replacing the BBC Trust with a unitary board. The BBC remains concerned about proposals for the government to appoint almost half the board members, which critics say is a direct attack on editorial independence.

Speaking at the unveiling of the BBC’s annual report on Tuesday, Tony Hall, the director general, said that it was a “good, hard, tough, difficult negotiation” that he hoped would not change. “I have no reason to think Theresa May won’t think it’s right,” he said.

The government is still considering whether to sell off a stake in Channel 4. Proposals to completely privatise the state-owned, commercially funded broadcaster were scrapped earlier this year.

May will also have to decide whether to launch the second part of the Leveson inquiry into press ethics, which was put on hold amid criminal inquiries over phone hacking. The government has shown no desire to open the planned second inquiry, which was to have looked at the ties between police and journalists, despite calls from campaigning groups and Labour. Mark Sweney

One key question is whether the UK will ratify last year’s historic Paris climate deal as part of the EU, or after it leaves. There is also uncertainty over political backing for new nuclear at Hinkley Point, whether the autumn statement will yield clarity for low carbon energy subsidies beyond 2020, and whether a plan on how to meet carbon targets will be published before the year is out, as previously planned.

In October, May’s environment secretary will face a legal challenge at the high court over what campaigners say is an inadequate government plan to clean up the illegally high air pollution in many British cities. The environment department will also have to grapple with the future of support for farmers who currently make more than half of their income from EU subsidies. Adam Vaughan

Justice

The most controversial item on the justice agenda is the repeatedly delayed bill of rights aimed at redefining the UK’s relationship with the European Court of Human Rights in Strasbourg.

Once the leadership campaign commenced, however, May swiftly reversed her position, acknowledging that there was no chance of a parliamentary majority to achieve that aim.

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She may also have calculated she needed the support of liberal-minded Tories. The need to concentrate on EU negotiations suggests the long-anticipated bill of rights may drop to the back of any legislative queue, or even be quietly forgotten.

The justice budget has been subject to deep cuts under austerity policies, forcing ministers to raise court and tribunal fees to such high levels that many lawyers warn they restrict access to justice. May could be sensitive to jibes that the British courts are open to wealthy Russian oligarchs pursuing legal vendettas but closed to ordinary UK citizens. Owen Bowcott

Transport

Cameron’s early departure means he has concluded his political tightrope act of appearing to take action on airport expansion without reneging on his personal pledge to never build a third runway at Heathrow. More than a year has already elapsed without a proper governmental response since the Airports Commission recommended the runway be built. But May, a presumed but not explicit opponent of Heathrow expansion , would be expected to take a decision in the autumn.

HS2 is effectively in process, but recent costcutting measures such as deciding not to build a new Sheffield station, and possible budgetary pressures ahead, suggest the high-speed project could be further trimmed – or in campaigners’ highest hopes, scrapped – before its full parliamentary approval this winter.

Furious commuters, many in solid Tory constituencies, have demanded action over the disastrous performance of Southern Rail. Emergency measures such as breaking up the franchise or bringing the line back under state control would be counterintuitive for Conservatives but May has already shown she would countenance other ideas when popular or pragmatic. Gwyn Topham